‎for any archimedean$f$-ring $a$ with unit in whichbreak$awedge‎‎(1-a)leq 0$ for all $ain a$‎, ‎the following are shown to be‎‎equivalent‎:‎‎1‎. ‎$a$ is isomorphic to the $l$-ring ${mathfrak z}l$ of all‎‎integer-valued continuous functions on some frame $l$‎. 2‎. ‎$a$ is a homomorphic image of the $l$-ring $c_{bbb z}(x)$‎‎of all integer-valued continuous functions‎, ‎in the usual sense‎,‎on som...

For a real number $\alpha $ and natural $N$, the Sudler product is defined by $P_N(\alpha ) = \prod _{r=1}^{N} 2 \lvert \sin (\pi r\alpha )\rvert $. Denoting $F_n$ $n$th Fibonacci $\phi Golden Ratio, we show that for $F

in this work, byemploying the leggett-williams fixed point theorem, we study theexistence of at least three positive solutions of boundary valueproblems for system of third-order ordinary differential equationswith $(p_1,p_2,ldots,p_n)$-laplacianbegin{eqnarray*}left { begin{array}{ll} (phi_{p_i}(u_i''(t)))'  +  a_i(t) f_i(t,u_1(t), u_2(t), ldots, u_n(t)) =0 hspace{1cm} 0  leq t leq 1, alpha_i u...

In this paper we investigate analytic properties of sextet polynomials hexagonal systems. For the pyrene chains, show that zeros $$P_n(x)$$ are real, located in open interval $$(-3-2\sqrt{2},-3+2\sqrt{2})$$ and dense corresponding closed interval. We also coefficients symmetric, unimodal, log-concave, asymptotically normal. general systems, real all $$(-\infty ,0]$$ , conjecture every polynomia...